Question

Differentiate w.r.t.x - y=e^(xloga) + e^(alogx) + e^(aloga)

Answer 1

Differentiate w.r.t.x

$$y = e ^ {x \log a} + e ^ {a \log x} + e ^ {a \log a}$$

**Solution:**

We'll use next rules

1. $\left(e^{f(x)}\right)' = e^{f(x)} \cdot f'(x)$

2. $(\log x)' = \frac{1}{x \ln 10}$

3. $(f(x) + g(x))' = f'(x) + g'(x)$

4. $(cx)' = c$ where $c = \text{const}$

5. $c' = 0$ if $c = \text{const}$ .

Denote

$$y ^ {\prime} = \frac {d y}{d x}.$$

Because $a = \text{const}$ and $\log a = \text{const}$ then

$$\begin{array}{l} y ^ {\prime} = \left(e ^ {x \log a} + e ^ {a \log x} + e ^ {a \log a}\right) ^ {\prime} = \left(e ^ {x \log a}\right) ^ {\prime} + \left(e ^ {a \log x}\right) ^ {\prime} + \left(e ^ {a \log a}\right) ^ {\prime} = \\ = e ^ {x \log a} \cdot (x \log a) ^ {\prime} + e ^ {a \log x} \cdot (a \log x) ^ {\prime} + 0 = \log a e ^ {x \log a} + \\ + \frac {a}{x \ln 1 0} e ^ {a \log x}. \\ \end{array}$$

Answer:

$$y ^ {\prime} = \log a e ^ {x \log a} + \frac {a}{x \ln 1 0} e ^ {a \log x}$$